There are lots of reasons active managers don't beat a passive index: transaction costs and commissions, taxes, market impact, salaries and costs of researchers, the fees the managers charge. You could also bring skill differences into it.

But suppose NONE of those things are in play, and it's a pure frictionless zero-sum trading game, (so no costs/fees/taxes whatsoever, no market impact, equal skill) with the active managers trading amongst themselves, and in aggregate, exactly holding some passive index, so that on average they always exactly track that index. At any point in time there will be "winners" who are beating the index, and "losers" who are trailing the index. Some people may find it intuitively obvious that there should be roughly equal numbers of winners and losers, but that intuition would be totally wrong. In fact the overwhelming majority would be losers. Let's see why.

First a simple example. Consider 1024 players starting with $1 each, and in one round they pair off for a winner take all coin flip. So after one round 512 have $2 (the rest have zero), then after round 2, 256 have $4, ..., finally after round 10, there is 1 winner with $1024 and 1023 losers with $0, but the average holding is always $1, it is a zero-sum game, every bet has zero expectation.

A second example, suppose a large number of players start with an equal stake. In each round each player bets 10% of their current stake on a coin flip, so their stake is multiplied by 1.1 or 0.9 with 50% chance of each. After the flip, the arithmetic mean of the stake is (1.1+0.9)/2=1 times their previous stake, but the geometric mean is sqrt(1.1*0.9)=0.995 (approx). After each round, the expected (arithmetic mean) stake is unchanged, but the geometric mean (and median) of the stakes drops by 0.5%, and the distribution of stakes tends towards lognormal with this geometric mean (and median). The median player is a loser. Only players who win at least 52.5% of flips (solve (1.1)^p(0.9)^(1-p)=1) are winners, and for a larger number of flips, the winners are a shrinking minority. (This is set up as if players are betting against a banker, but you could modify it to be players are betting against each other so that the total of all stakes is unchanged, and it's a zero-sum game with almost all losing in the long run.)

Now for trading. Suppose a pool of 10,000 passive investors start with $100,000 each of a total stock market index, and just hold it. (For simplicity, suppose no dividends). Suppose a pool of 10,000 active investors start with the same stocks, but then proceed to continually trade individual stocks amongst themselves. Suppose the whole process is absolutely frictionless and cost- and tax- free (with no skill differences, no market impact etc). In other words, eliminate all the usual reasons for active trading mostly losing. Now the passive and active pool each start with a total of $1B stock, which over time goes up and down, but at all times, the passive and active pools are exactly tied in aggregate, and also the individual passive investors are always exactly tied with each other, but the individual active investors will have differing balances depending on the performance of their chosen stocks. At any time, each active investor is a winner or loser, depending on whether they hold more or less than the average (arithmetic mean), while every passive investor is exactly average at all times.

The active investors more concentrated bets are like the coin flip examples, where the average (arithmetic mean) of their holdings always exactly tied with the passive investors, but the geometric mean (and median) is increasingly lower as time goes on, and the distribution of the active investors's balances tends towards lognormal with this geometric mean (and median), so that an increasing majority of the active investors are losers. In the long run, amongst the active investors there will be 9,999 losers and 1 winner (who was purely lucky, as there is assumed no skill difference).

Of course, the argument could be made more rigorous with a bunch more math, but it's the usual stuff with arithmetic and geometric means, and multiplicative version of Central Limit Theorem (and even if mCLT isn't fully applicable, the conclusion that most active investors would be losers is solid).

There are a bunch of real life reasons that cause most active funds to trail the index. But even if all those reason were eliminated, leaving a pure zero-sum trading game, it would still be that after enough time, almost all active traders are below average. If your intuition was that there should be roughly equal numbers of winners and losers, then your intuition is totally wrong. There are few winners and almost all losers.

I don't think it was mentioned, but another reason why active managers struggle to outperform passive over long periods is because most of them aren't fully invested. They hold more cash than their passive counterparts.

I don't think it was mentioned, but another reason why active managers struggle to outperform passive over long periods is because most of them aren't fully invested. They hold more cash than their passive counterparts.

There are a bunch of real life reasons that cause most active funds to trail the index. But even if all those reason were eliminated, leaving a pure zero-sum trading game, it would still be that after enough time, almost all active traders are below average. If your intuition was that there should be roughly equal numbers of winners and losers, then your intuition is totally wrong. There are few winners and almost all losers.

Thank you for an interesting post, I believe this should also be emphasized when comparing index and active investing.

Last edited by buylowbuyhigh on Wed Dec 05, 2018 5:29 am, edited 2 times in total.

I don't think it was mentioned, but another reason why active managers struggle to outperform passive over long periods is because most of them aren't fully invested. They hold more cash than their passive counterparts.

I think the point is that a big majority of active managers fall behind their collective average or mean return over time. Holding cash, like fees and costs, only pushes this average down from the market return. Example could be: market return 7%, average index fund 6.9%, median index fund 6.9%, average active fund 6%, median active fund 5%.

I don't think it was mentioned, but another reason why active managers struggle to outperform passive over long periods is because most of them aren't fully invested. They hold more cash than their passive counterparts.

Sure. So add that to the list of typical reasons that cause most active funds to trail the index.

What I'm saying in this thread is suppose that NONE of these reasons apply, and you have a pure frictionless zero-sum trading game, with the active managers trading amongst themselves, so they are always guaranteed to be tied on average with the passive investors. EVEN THEN, when we have removed all the usual headwinds against active investing, the expected outcome in the long run is that almost all active traders will be below average.

Interesting point. The payoff distribution gets right skewed because the first round winners will place larger bets in the second round (because they have more money) and the first round losers will place smaller bets (because they have less money). After a few rounds, you'll have a few mega winners (those who placed larger and larger winning bets) but you won't have mega losers because the losers keep trimming their bets.

A nice point to add to the Arithmetic of Active Management. Not only is the the excess return zero before costs (and negative after costs), it's also right skewed, which is not an attractive payoff.

There are a bunch of real life reasons that cause most active funds to trail the index. But even if all those reason were eliminated, leaving a pure zero-sum trading game, it would still be that after enough time, almost all active traders are below average. If your intuition was that there should be roughly equal numbers of winners and losers, then your intuition is totally wrong. There are few winners and almost all losers.

Thank you for an interesting post, I believe this should also be emphasized when comparing index and active investing.

Same is true for sports betting, in reality, it's a 50/50 coin toss against the spread (assuming you take out a push). Vegas charges you 10% to use their services. So you would need to win 11 out of 20 bets just to break even against the spread. This 10% aside, most "experts" don't even come close to 500 wins over the course of 1000 bets, only the handicappers come close. Why is this the case if it's truly a coin toss, I don't know, but it is. I haven't come across anyone who wins 3 out of 5 or 60%, creating the 5% spread that would make it profitable.

Are people here still tempted by active funds? I thought that most Bogleheads would prefer index funds to active funds without having to be convinced.

Probably the biggest drawback to "active" funds are the high costs. Bogleheads do strongly support index funds predominantly, however Vanguard has some (relatively) low-cost actively-managed funds such as Wellesley/Wellington that have done very well for many past decades.

[quote]The active investors more concentrated bets are like the coin flip examples, where the average (arithmetic mean) of their holdings always exactly tied with the passive investors, but the geometric mean (and median) is increasingly lower as time goes on, and the distribution of the active investors's balances tends towards lognormal with this geometric mean (and median), so that an increasing majority of the active investors are losers. In the long run, amongst the active investors there will be 9,999 losers and 1 winner (who was purely lucky, as there is assumed no skill difference).
[/quote]

This assumes performance persistence with one lucky winner consistently ending up on the positive side.
If the trading were random, that would be possible, but highly unlikely. The likely outcome of random trading would be that the active investors end up with the same average wealth as the passive investors. The active investors might have a wider range of outcomes relative to the passive investors. But there is no reason to assume that all but one of the active investors ends up a loser.

Try it this way. After each round the likelihood of winning for each individual investor is the same across time and across the cross section of investors. Long runs of positive or negative returns relative to the market are possible, but unlikely. The likely outcome is an exact match of the market returns.

Not a reason to engage in active management. But no reason to claim that there would be only one winner.

We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either |
--Swedroe |
We assume that markets are efficient, that prices are right |
--Fama

Interesting point. The payoff distribution gets right skewed because the first round winners will place larger bets in the second round (because they have more money) and the first round losers will place smaller bets (because they have less money). After a few rounds, you'll have a few mega winners (those who placed larger and larger winning bets) but you won't have mega losers because the losers keep trimming their bets.

A nice point to add to the Arithmetic of Active Management. Not only is the the excess return zero before costs (and negative after costs), it's also right skewed, which is not an attractive payoff.

Yes, you link the classic Sharpe argument that if the active traders hold "the market" in aggregate, but that costs are draining some wealth away, then on average they must trail the market.

My argument is that even if there are zero costs so they exactly match the market on average, the losers will still be the overwhelming majority.

The active investors more concentrated bets are like the coin flip examples, where the average (arithmetic mean) of their holdings always exactly tied with the passive investors, but the geometric mean (and median) is increasingly lower as time goes on, and the distribution of the active investors's balances tends towards lognormal with this geometric mean (and median), so that an increasing majority of the active investors are losers. In the long run, amongst the active investors there will be 9,999 losers and 1 winner (who was purely lucky, as there is assumed no skill difference).

This assumes performance persistence with one lucky winner consistently ending up on the positive side.
If the trading were random, that would be possible, but highly unlikely. The likely outcome of random trading would be that the active investors end up with the same average wealth as the passive investors. The active investors might have a wider range of outcomes relative to the passive investors. But there is no reason to assume that all but one of the active investors ends up a loser.

Try it this way. After each round the likelihood of winning for each individual investor is the same across time and across the cross section of investors. Long runs of positive or negative returns relative to the market are possible, but unlikely. The likely outcome is an exact match of the market returns.

Not a reason to engage in active management. But no reason to claim that there would be only one winner.

No, your intuition is definitely wrong. (One reason for this thread is to present something which is counterintuitive to many people, though there are many others who know the theory.) Firstly the assumption is there are no costs, so the active traders are absolutely guaranteed to have exactly the same same average wealth as the passives. Furthermore it is assumed the active traders are equally skilled, so their results are essentially pure luck.

To be more concrete, assume each active trader, each day, randomly bets on part of the market (such that the active traders always hold the same, in aggregate, as the passives, i.e. the actives are trading amongst themselves). Or set it up in any way that the multiplicative version of Central Limit Theorem holds, at least approximately. With mCLT (no need to assume anything about the distribution of returns of individual stocks) the distribution of the active traders' balances tends towards lognormal, where the arithmetic mean exactly matches that of the passives, while the geometric mean (and median) falls increasingly behind, as time elapses, so that fewer and fewer actives are winners.

Then the conclusion is indeed that after sufficient time, it is almost certain that amongst the actives, there will be exactly 1 winner (above average) and the rest are all losers (below average), while the passives are all exactly average (hence tied for 2nd place). This is a rigorous conclusion, not an assumption.

This assumes performance persistence with one lucky winner consistently ending up on the positive side.
If the trading were random, that would be possible, but highly unlikely. The likely outcome of random trading would be that the active investors end up with the same average wealth as the passive investors.

In large samples, unlikely events happen a lot. The average of actives must be equal to the average of passives (before costs). The likely outcome for each active investor is worse (before costs). Median is not mean.

Once you understand the math, the hardest thing for people to accept is the randomness of the situation. Everyone thinks that they are special due to their own unique insight and skill.

When I started investing little over five years ago, I first went with active funds and briefly with individual stocks, trailing the basic indices by ~5% per year. I had no idea what I was doing, but it's nice to think I was also unlucky. At least I'm lucky for being born to a time when index funds are already available for everyone (not just in the US).

Once you understand the math, the hardest thing for people to accept is the randomness of the situation. Everyone thinks that they are special due to their own unique insight and skill.

I don't know that what I said is literally a case of "gambler's ruin", but it is definitely in the same general area of probability and statistics, and was surely well understood by the 1600s (while things like inequality of arithmetic and geometric means would be literally ancient). I'm sure it was explicitly applied to investing by writers many decades ago (hence "diversification is the only free lunch" etc).

In my example the "gamblers" (active traders) don't generally go totally broke, but mostly fall further and further behind the index. E.g. suppose each day, each active trader randomly selects one sector to invest in (subject to the constraint that in aggregate, the active traders always hold the total market, and so are always tying the passives in aggregate). So each day, each active trader has exactly the same expected (arithmetic mean) return as the passives, but the results are more dispersed, so the geometric mean is lower. Iterate this for many days, and an ever increasing majority of the active investors trail the market. (The market could be up, but most actives get less than what the passives do.)

I believe the OP's artificial game is an incorrect analogy and doesn't truly explain why most active managers under perform even if you backed out all costs. The stock market is not a random game with an arithmetic/geometric mean issue and volatility decay. The stock market is a reflection of the economy and the economy is naturally right skewed with with a few big winners and lots of losers. Every year, 10 stocks account for 50% of the stock market's gains (or whatever), so if you're an active manager, there's a good chance you are underweight those necessary winners since there are so few.

I've seen the arithmetic/geometric mean argument before but there's no reason to think that that is the distribution that exists in the real world.

The active investors more concentrated bets are like the coin flip examples, where the average (arithmetic mean) of their holdings always exactly tied with the passive investors, but the geometric mean (and median) is increasingly lower as time goes on, and the distribution of the active investors's balances tends towards lognormal with this geometric mean (and median), so that an increasing majority of the active investors are losers. In the long run, amongst the active investors there will be 9,999 losers and 1 winner (who was purely lucky, as there is assumed no skill difference).

Of course, the argument could be made more rigorous with a bunch more math, but it's the usual stuff with arithmetic and geometric means, and multiplicative version of Central Limit Theorem (and even if mCLT isn't fully applicable, the conclusion that most active investors would be losers is solid).

That's an interesting argument. Let me reword it slightly. You are asserting that, in a frictionless/costless environment, the average (mean) return is the same between passive and active investors (usual Sharpe argument), but that the standard-deviation of active investors returns is higher because of 'more concentrated bets'. And then consequently, the CAGR will suffer. Well, one could point out that value investors (including active funds like Wellington) might beg to differ, but this is probably besides your point. Yes, I can see your argument. Seems fair enough. Thanks for sharing.

The fact that trading is zero-sum doesn't inherently mean it's not possible to identify those on the outperforming end of the scale. And the fact that the average fund underperforms an index should only scare someone if they're mandated to select an average (or worse) active fund.

Also, I haven't seen an updated article from Vanguard, but a few years back, they posted their own article that showed their own actives are outperforming their index funds, and it was by about 30-40bp if i recall, NET of fees. Just luck to outperform in aggregrate despite that many funds? I don't think so.

Just my opinion, but one of Boglehead's strongest viewpoint is the fight against high costs, and should focus much more on that instead of Active vs. Passive. Active just so happens to be mostly "bad" because it's usually (too) expensive. But there are exceptions on both sides of the equation (e.g. expensive "index" funds, and cheap (consistently outperforming) active funds).

If you buy the right funds, you can sometimes get an strongly performing active fund for an extra 15bp or so, vs. a comparable index fund. 15bp is pretty cheap for an upgrade from a "trained-monkey".

I believe the OP's artificial game is an incorrect analogy and doesn't truly explain why most active managers under perform even if you backed out all costs. The stock market is not a random game with an arithmetic/geometric mean issue and volatility decay. The stock market is a reflection of the economy and the economy is naturally right skewed with with a few big winners and lots of losers. Every year, 10 stocks account for 50% of the stock market's gains (or whatever), so if you're an active manager, there's a good chance you are underweight those necessary winners since there are so few.

I've seen the arithmetic/geometric mean argument before but there's no reason to think that that is the distribution that exists in the real world.

Firstly, arithmetic geometric mean inequality is a theorem. It's never not true. And it has lots of powerful implications in finance.

Secondly, let me spell out the logic. There are a whole bunch of legitimate real world reasons (e.g. costs, skill differences, etc) that lead to the conclusion that most actives trail the market. The logical structure of my argument is that EVEN IF those assumptions don't hold, the conclusion STILL holds. Those assumptions aren't needed to nevertheless reach the conclusion.

So while it may be true that empirically returns are found to be right skewed (and this would in fact be in part due to reasons in the OP), the conclusion that most actives trail the market can STILL be concluded EVEN IF we don't assume returns right skewed. That assumption is not needed to nevertheless reach the conclusion.

Consider a bet where you gain 1% (stake multiplied by 1.01) with probability 99%, and lose 99% (stake multiplied by 0.01) with probability 1%. That's a fair, even money bet, which the gambler wins most of the time (and it's strongly LEFT skewed). But if you repeatedly put your whole stake into this bet (letting it ride), then in the long run, the results are distributed lognormally, which is right skewed, and you almost certainly lose. A typical median 100 bets have your bankroll multiplied by (1.01)^99*(0.01)^1=0.02678, so the median result is to lose 97.322% of your stake over the course of 100 bets.

The active investors more concentrated bets are like the coin flip examples, where the average (arithmetic mean) of their holdings always exactly tied with the passive investors, but the geometric mean (and median) is increasingly lower as time goes on, and the distribution of the active investors's balances tends towards lognormal with this geometric mean (and median), so that an increasing majority of the active investors are losers. In the long run, amongst the active investors there will be 9,999 losers and 1 winner (who was purely lucky, as there is assumed no skill difference).

Of course, the argument could be made more rigorous with a bunch more math, but it's the usual stuff with arithmetic and geometric means, and multiplicative version of Central Limit Theorem (and even if mCLT isn't fully applicable, the conclusion that most active investors would be losers is solid).

That's an interesting argument. Let me reword it slightly. You are asserting that, in a frictionless/costless environment, the average (mean) return is the same between passive and active investors (usual Sharpe argument), but that the standard-deviation of active investors returns is higher because of 'more concentrated bets'. And then consequently, the CAGR will suffer. Well, one could point out that value investors (including active funds like Wellington) might beg to differ, but this is probably besides your point. Yes, I can see your argument. Seems fair enough. Thanks for sharing.

Yes your rewording is totally a legitimate way of looking at it.

And although I am stripping away almost all assumptions, I do need to assume something about the actives so that multiplicative version of Central Limit Theorem applies, and one way to get that is to suppose these actives are periodically changing holdings (randomly) so that over time they hold the market on average, just different portions at different times. The conclusions won't necessarily follow if actives just pick a segment of the market and hold it.

The fact that trading is zero-sum doesn't inherently mean it's not possible to identify those on the outperforming end of the scale. And the fact that the average fund underperforms an index should only scare someone if they're mandated to select an average (or worse) active fund. ..snip.. If you buy the right funds, you can sometimes get an strongly performing active fund for an extra 15bp or so, vs. a comparable index fund. 15bp is pretty cheap for an upgrade from a "trained-monkey".

Yes, I don't dispute that skill could beat the market. I'm just not using that assumption.

Then the conclusion is indeed that after sufficient time, it is almost certain that amongst the actives, there will be exactly 1 winner (above average) and the rest are all losers (below average), while the passives are all exactly average (hence tied for 2nd place). This is a rigorous conclusion, not an assumption.

Then show it rigorously. What you have stated does not lead to that conclusion. Others are adding in assumptions about winners behaving differently than losers after each days trading, about some people going to ruin, or other conditions.

But to your basic point of a tiny number of winners, perhaps 1.

There are a large number of active investors.
Each active investor starts by holding the entire market.
On day one half the active investors continue to hold the entire market except they sell a set number of shares of stock A, out of the 4,000 stocks in the market. At the same time, they buy an equal dollar amount of stock B. The following day, they all reverse this process. Those who sold A, now buy it. Those who bought B now sell it.

Over time, the diverging fortunes of stocks A and B, and the days on which one would be best off having held one or the other, would lead to one set of active investors having more money than the other. How much would depend on the volatility of those two stocks and how large were the bets the investors placed on them. If the active investors deviated from market weights by a large amount, greatly underweighting or overweighting their daily stock and if the dollar amounts they devoted to this were large, then the difference between, by luck, being more in A on its good days and more in B on its good days might end up as a substantial amount of money.

On the other hand, if A and B had started out as the two smallest stocks in the market, stayed the two smallest stocks in the market throughout the test period, displayed very little volatility, maintaining the same value all day long, year in and year out and the active investors never went beyond 99% or 101% of market weight, then one could run this forever with no reason to expect the lucky investors to be much better off than the unlucky ones.

My prediction:
Since almost their entire portfolio was and always remains the market (here 3,998 out of 4,000 stocks), the only way to go to zero would be if the entire market goes to zero along with at least one of stock A or stock B. Then the group that had an extra (above market) holding in the stock that still had value when all the other 3,999 stocks went to zero would have the residual value of their single stock. But even here, if, say, stock A has value, and everyone, active and passive, holds A, then how do they go to zero? So no one will go to zero unless the entire market does so.

Here you have daily trading, but no one goes to zero while the stock market overall has some value. One could do the trades, no transaction costs, millions of times a day and get the same result.

The creation of a small set of winners, perhaps down to one winner, cannot be a simple consequence of frictionless trading, no matter how much of it takes place. There has to be something else presumed to be going on to get there.

If after some long period of time, there will be only one winner? Why is that?

This calls for a little math to demonstrate it.

Last edited by afan on Thu Dec 06, 2018 4:14 pm, edited 2 times in total.

We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either |
--Swedroe |
We assume that markets are efficient, that prices are right |
--Fama

Yes, I don't dispute that skill could beat the market. I'm just not using that assumption.

I'm not understanding your brief response here, but I do think we're disagreeing, for reasons I've previously stated. In summary, if I'm only paying an extra 10-15 basis points for active management in a fund that I've researched with a long track record, that's going to be held in a tax-advantaged account (read turnover doesn't cost me taxes), and who's fund manager identified a style that I'm more comfortable with than market cap weighted (e.g. a preference for Value stock, instead of market-cap weighted which means that's those stocks that did better in the past are purchased in larger amounts (which seems backwards to me as to what you should do)), then I think active could be a good choice in that scenario. And FWIW, so does Vanguard.

Quite frankly, I just like knowing an actual human is at the helm assessing whatever the current situation, such as like the volatile times we're in now. I just can't believe no one has any idea at all about what to do. For one, it's not possible for me to believe that because I can't unlearn my knowledge of Warren Buffett. Yeah, I know a famous guy we admire once said no one knows nothing, yet that same man paired down his stock holdings massively in the late 90s due to the current investing climate. And he was plenty older than I am now, when he did that.

But do the overwhelming majority of active funds lose? I imagine that's absolutely true. But last I had heard, the average active fund also charges some 130 basis points. Of course those are going to lose.

Just my opinion, but one of Boglehead's strongest viewpoint is the fight against high costs, and should focus much more on that instead of Active vs. Passive. Active just so happens to be mostly "bad" because it's usually (too) expensive. But there are exceptions on both sides of the equation (e.g. expensive "index" funds, and cheap (consistently outperforming) active funds).

If you separate Bogle and Vanguard from the Bogleheads for a minute you'll see that they have been winning for decades on the Active side and for all intents and purpose restructured the entire Mutual Fund industry while they were at it and still came out with a nearly unbeatable hand (contra index funds where Vanguard has real hard to be beat price competition).

I haven't seen recent numbers, but while almost everyone has been seeing outflows from Active funds, Vanguard outflows have been relatively mild because their active fees are literally 1/3 of most of the competition. So they often start out with 50bp lead and end up at the top quartile of the pile almost every year in the style boxes and nearly 100% of the time over longer periods like a decade...

Jack used to push a model of I think it was 65% index 35% well chosen actives and I followed that for a long time, he folded on that probably 15 years ago and basically preaches S&P500 / TSM all the way.

The problem is it requires more maintenance than 3 or 4 fund portfolio with very little upside potential (granted some).

So the BH philosophy said, what the heck, go all the way. the price of simplicity is very very low. The value of simplicity is broad and can be quite liberating. I finally fully embraced that philosophy in the last 2 years and am very very happy i did so. My spreadsheets have gotten a lot simpler.... for one thing I don't care about style boxes anymore

Just my opinion, but one of Boglehead's strongest viewpoint is the fight against high costs, and should focus much more on that instead of Active vs. Passive. Active just so happens to be mostly "bad" because it's usually (too) expensive. But there are exceptions on both sides of the equation (e.g. expensive "index" funds, and cheap (consistently outperforming) active funds).

If you separate Bogle and Vanguard from the Bogleheads for a minute you'll see that they have been winning for decades on the Active side and for all intents and purpose restructured the entire Mutual Fund industry while they were at it and still came out with a nearly unbeatable hand (contra index funds where Vanguard has real hard to be beat price competition).

I haven't seen recent numbers, but while almost everyone has been seeing outflows from Active funds, Vanguard outflows have been relatively mild because their active fees are literally 1/3 of most of the competition. So they often start out with 50bp lead and end up at the top quartile of the pile almost every year in the style boxes and nearly 100% of the time over longer periods like a decade...

Jack used to push a model of I think it was 65% index 35% well chosen actives and I followed that for a long time, he folded on that probably 15 years ago and basically preaches S&P500 / TSM all the way.

The problem is it requires more maintenance than 3 or 4 fund portfolio with very little upside potential (granted some).

So the BH philosophy said, what the heck, go all the way. the price of simplicity is very very low. The value of simplicity is broad and can be quite liberating. I finally fully embraced that philosophy in the last 2 years and am very very happy i did so. My spreadsheets have gotten a lot simpler.... for one thing I don't care about style boxes anymore

I appreciate many of the points you made there, but you closed with a bit of a false dichotomy. I could propose a very simple 3-4 fund portfolio that's 100% active, 100% still low cost (certainly in comparison to the average active fund), and is plenty diversified. Active might have its issues, but building a simple portfolio with 100% actives is not among that list. I don't really want market-cap weighted anything, for reasons I previously stated. For example, have mostly Apple stock because Apple to date has been/was/(read: past tense) a great stock in the past, and is bid up to the stratosphere? I think I'll pass on that.

But we agree on making it simple as possible. Total 100% agreement on that point.

I believe the OP's artificial game is an incorrect analogy and doesn't truly explain why most active managers under perform even if you backed out all costs. The stock market is not a random game with an arithmetic/geometric mean issue and volatility decay. The stock market is a reflection of the economy and the economy is naturally right skewed with with a few big winners and lots of losers. Every year, 10 stocks account for 50% of the stock market's gains (or whatever), so if you're an active manager, there's a good chance you are underweight those necessary winners since there are so few.

I've seen the arithmetic/geometric mean argument before but there's no reason to think that that is the distribution that exists in the real world.

Yes, this was exactly my first line of thinking when reading the OP. The analogy is clearly dubious because a typical distribution of returns would definitely not look anything like the dice game being described. Furthermore, one could have a distribution of returns that compensates for the arithmetic/geometric issue being identified (heck, there are *some* winners in this story). But the analogy being imperfect doesn't defeat the core argument in itself. And when I reworded the OP's argument based on arithmetic returns and (higher expected) standard deviation, this seemed to make much more sense. Something still nags me about it though... Will think more about it tonight.

(@ some comments further up) I am talking about a simple pure math model. You can always debate its applicability to the real world, but the math stands.

Also the conclusion (most actives trail) does require some assumptions about the behavior of the actives:

What I call "actives" are modeled like equally skilled dart throwing monkeys who each day (or whatever time interval) randomly (and independently each day) choose part of the market to invest in (subject to the constraint that in aggregate, the active traders always hold the total market, and so are always tying the passives in aggregate). So each day, each active trader has exactly the same expected (arithmetic mean) return as the passives, but the results are more dispersed, so the geometric mean is lower. Iterate this for many days, and an ever increasing majority of the active investors trail the market.

Basically the probability distribution of the balance of an active is the balance of a passive multiplied by a product of independent random variables X_1 X_2 X_3 ... X_n each with expected value 1 but geometric mean less than 1, so after sufficient interations (large n) the product approaches lognormal^* and the probability of a particular active beating a passive approaches zero. The independence of the random variables is built into the presumed active trader behavior, and makes no assumptions about the statistics of investment returns.

The conclusions should still follow after somewhat loosening the assumptions about the behavior of the actives.

It's easy to contrive examples where the conclusion doesn't follow, if the behavior of active traders is sufficiently different from what is described here. If you want a counterexample to "If P then Q", then you not going to get it by denying P (but can deny Q by itself if you deny P).

The easiest way to understand what is being said, is to understand the math purely abstractly, and then see how that translates to the investing scenario.

I believe the OP's artificial game is an incorrect analogy and doesn't truly explain why most active managers under perform even if you backed out all costs. The stock market is not a random game with an arithmetic/geometric mean issue and volatility decay. The stock market is a reflection of the economy and the economy is naturally right skewed with with a few big winners and lots of losers. Every year, 10 stocks account for 50% of the stock market's gains (or whatever), so if you're an active manager, there's a good chance you are underweight those necessary winners since there are so few.

I've seen the arithmetic/geometric mean argument before but there's no reason to think that that is the distribution that exists in the real world.

Yes, this was exactly my first line of thinking when reading the OP. The analogy is clearly dubious because a typical distribution of returns would definitely not look anything like the dice game being described. Furthermore, one could have a distribution of returns that compensates for the arithmetic/geometric issue being identified (heck, there are *some* winners in this story). But the analogy being imperfect doesn't defeat the core argument in itself. And when I reworded the OP's argument based on arithmetic returns and (higher expected) standard deviation, this seemed to make much more sense. Something still nags me about it though... Will think more about it tonight.

Nothing is being assumed about the statistics of investment returns, because instead it is being assumed that the "actives" are choosing randomly.

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

The crux (and Achilles' heal) of the argument is that at some point it needs to be able to use the multiplicative version of Central Limit Theorem so the scenario needs to be set up so that mCLT is applicable, or at least approximately so.

Anyway, it's worth understanding the abstract math model on its own merits (walk before run). Then figure when/where it's applicable to the messy concrete real world.

I believe the OP's artificial game is an incorrect analogy and doesn't truly explain why most active managers under perform even if you backed out all costs. The stock market is not a random game with an arithmetic/geometric mean issue and volatility decay. The stock market is a reflection of the economy and the economy is naturally right skewed with with a few big winners and lots of losers. Every year, 10 stocks account for 50% of the stock market's gains (or whatever), so if you're an active manager, there's a good chance you are underweight those necessary winners since there are so few.

I've seen the arithmetic/geometric mean argument before but there's no reason to think that that is the distribution that exists in the real world.

Yes, this was exactly my first line of thinking when reading the OP. The analogy is clearly dubious because a typical distribution of returns would definitely not look anything like the dice game being described. Furthermore, one could have a distribution of returns that compensates for the arithmetic/geometric issue being identified (heck, there are *some* winners in this story). But the analogy being imperfect doesn't defeat the core argument in itself. And when I reworded the OP's argument based on arithmetic returns and (higher expected) standard deviation, this seemed to make much more sense. Something still nags me about it though... Will think more about it tonight.

Nothing is being assumed about the statistics of investment returns, because instead it is being assumed that the "actives" are choosing randomly.

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

The crux (and Achilles' heal) of the argument is that at some point it needs to be able to use the multiplicative version of Central Limit Theorem so the scenario needs to be set up so that mCLT is applicable, or at least approximately so.

Anyway, it's worth understanding the abstract math model on its own merits (walk before run). Then figure when/where it's applicable to the messy concrete real world.

Your game is a strawman. Basically you create a game that looks like the stock market (but is actually quite different from the stock market) then you go on to "prove" that playing that game loses money and therefore playing the stock market loses money. Just because your math is correct, it doesn't mean your comparison of the betting game to the stock market is an appropriate comparison.

First of all, your game doesn't explain why the index outperforms active managers. It instead says that if you play that particular game you go to zero eventually. But there's no reason to think the game you made up represents the stock market, for example no one argues that being in the stock market leads you to zero the longer you are in it.

Your argument is the same one that a quant firm call INTECH uses to try and outperform. It was started by Robert Fernholz who introduced the phenomenon you're describing here which is often referred to as "stochastic portfolio theory". My main point is that I don't believe volatility decay is the fundamental reason why active managers underperform. If that was the case, you could actually use volatility decay to outperform the index, which is what INTECH attempts to do.

(@ some comments further up) I am talking about a simple pure math model. You can always debate its applicability to the real world, but the math stands.

Also the conclusion (most actives trail) does require some assumptions about the behavior of the actives:

What I call "actives" are modeled like equally skilled dart throwing monkeys who each day (or whatever time interval) randomly (and independently each day) choose part of the market to invest in (subject to the constraint that in aggregate, the active traders always hold the total market, and so are always tying the passives in aggregate). So each day, each active trader has exactly the same expected (arithmetic mean) return as the passives, but the results are more dispersed, so the geometric mean is lower. Iterate this for many days, and an ever increasing majority of the active investors trail the market.

Basically the probability distribution of the balance of an active is the balance of a passive multiplied by a product of independent random variables X_1 X_2 X_3 ... X_n each with expected value 1 but geometric mean less than 1, so after sufficient interations (large n) the product approaches lognormal^* and the probability of a particular active beating a passive approaches zero. The independence of the random variables is built into the presumed active trader behavior, and makes no assumptions about the statistics of investment returns.

The conclusions should still follow after somewhat loosening the assumptions about the behavior of the actives.

It's easy to contrive examples where the conclusion doesn't follow, if the behavior of active traders is sufficiently different from what is described here. If you want a counterexample to "If P then Q", then you not going to get it by denying P (but can deny Q by itself if you deny P).

The easiest way to understand what is being said, is to understand the math purely abstractly, and then see how that translates to the investing scenario.

I don't really think CLT assumptions have much to do with this.

Basically, the idea is that you have a finite Markov Chain with the state being defined as the N-tuple of the fortunes (say the fortunes are in dollars and we don't allow sub-dollar bets) of the N players. There is one absorbing state: When one player has all the money and everyone else has zero. If you force all the players to wager at least one dollar (or a variety of other assumptions would also serve) of their fortune at each step of the chain, then with probability one, the finite chain will eventually hit the absorbing state. The only way this doesn't happen is if a subset of the players collude and refuse to play one another (the Boglehead Philosophy).

Many (perhaps most) gamblers would lose to the house even if casinos had zero edge on their games, simply because most have poor bankroll management and take on too much risk pretty much guaranteeing eventual ruin / bankroll depletion when a bad sequence occurs, it is inevitable if too much risk is repeatedly taken.
The Kelly Criterion gives a mathematical formula to determine how much of the bankroll can be bet to maximize growth rate and eliminate the risk of ruin... It will also will tell you the amount to bet on a game with zero expected advantage(EV) is nothing.

Beating the market is a zero-sum game (you can't have a portfolio that achieves above market returns without one that underperforms). If we presume that markets are efficient (even though I don't think they are), then there is no positive expected value from trading, it's just trading different levels of risk/risk preferences. How much risk (relative to the market) the active trader is assuming can be measured through "active share" and seeing how much it deviates from the market return. Most mutual funds don't have very high active share, and the returns typically show a bell curve with the index being somewhere around the middle of the distribution. If you look at the tails (on either side) I'm sure you will find a lot more 'active share'. You would also likely see a lot more left-tail underperforming funds than outperforming funds on the right side of the distribution, and this can be explained by both high fees and by taking too much risk in the zero-sum trading game.
If markets are not efficient, it's still a zero-sum trading game, but it's even worse for traders with no information - if there are traders who do have information and asymmetrical advantages, then the zero information trader has negative expected value.
So in an efficient market (zero EV) Kelly Criterion would suggest you bet nothing on active risk.
In an in-efficient market where you have no special information (negative EV) Kelly Criterion would also suggest you bet nothing on active (actually it would suggest you bet on the guy who has the information, but you presumably don't know who that is).

"To achieve satisfactory investment results is easier than most people realize; to achieve superior results is harder than it looks." - Benjamin Graham

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

Yes, of course, but one could change some of the input parameters... Think to the sustained over-performance of Small-Cap Value compared to other strategies. The math theorem stays true, this is a more 'concentrated' investment, the standard-deviation was higher than the total-market returns, and yet it worked remarkably well in the past, definitely beating the market (crushing it, actually), and for more than a single winner. Why? Well, the arithmetic returns stayed consistently higher than the total market... Will that stay true in the future remains to be seen, but it certainly did work in the past, and not only in academic simulations, but also in real-life.

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

Yes, of course, but one could change some of the input parameters... Think to the sustained over-performance of Small-Cap Value compared to other strategies. The math theorem stays true, this is a more 'concentrated' investment, the standard-deviation was higher than the total-market returns, and yet it worked remarkably well in the past, definitely beating the market (crushing it, actually), and for more than a single winner. Why? Well, the arithmetic returns stayed consistently higher than the total market... Will that stay true in the future remains to be seen, but it certainly did work in the past, and not only in academic simulations, but also in real-life.

Anyway, it's worth understanding the abstract math model on its own merits (walk before run). Then figure when/where it's applicable to the messy concrete real world.

Yup, exactly, this is where the reasoning still nags me. It's an interesting discussion! Will reflect more upon it.

The math I said does NOT show that buying a concentrated part of the market and holding it is worse than holding the whole market. Obviously the concentrated bet could win. What is bad is repeatedly switching between different parts of the market. If you repeatedly do this randomly such that on average you held the market, then you almost certainly would have been better of holding the whole market.

Or if you repeatedly randomly switched between being 100% stock and 100% cash so that you were in each half the time, then you almost certainly would have been better of holding half of each the whole time.

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

Yes, of course, but one could change some of the input parameters... Think to the sustained over-performance of Small-Cap Value compared to other strategies. The math theorem stays true, this is a more 'concentrated' investment, the standard-deviation was higher than the total-market returns, and yet it worked remarkably well in the past, definitely beating the market (crushing it, actually), and for more than a single winner. Why? Well, the arithmetic returns stayed consistently higher than the total market... Will that stay true in the future remains to be seen, but it certainly did work in the past, and not only in academic simulations, but also in real-life.

Anyway, it's worth understanding the abstract math model on its own merits (walk before run). Then figure when/where it's applicable to the messy concrete real world.

Yup, exactly, this is where the reasoning still nags me. It's an interesting discussion! Will reflect more upon it.

The math I said does NOT show that buying a concentrated part of the market and holding it is worse than holding the whole market. Obviously the concentrated bet could win. What is bad is repeatedly switching between different parts of the market. If you repeatedly do this randomly such that on average you held the market, then you almost certainly would have been better of holding the whole market.

Or if you repeatedly randomly switched between being 100% stock and 100% cash so that you were in each half the time, then you almost certainly would have been better of holding half of each the whole time.

So then there's no point to any of your math. Who is using a stock picking method of "randomly" switching between different parts of the market, or hiring someone who's doing that? I think it could have gone without saying that probably isn't a very good method of investing, and hopefully no one is doing that. If I'm going to hire an active manager, I'm going to pay a small premium to do it, and I'm going to have good reason to believe he has a better than average chance of beating a market index long-term, net of fees.

^ I'm pretty sure that in practice, the conclusions hold much more widely, and you don't really need the assumption that are needed for a rigorous theoretical math argument.

As someone pointed out above, or see this threadviewtopic.php?f=10&t=248243
in practice, with actual data, returns themselves are right skewed. It's seems to be a kind of natural phenomenon. So it also shouldn't be surprising if it's similar for active traders, even if the analogy is rough.

A "below-mean" skewed return for most stocks is not necessarily a problem for an active manager. In a small portfolio (say under 100 stocks mutual fund), you only really need one or very few "future" "apples, amazons, or googles". The majority of the remaining stocks can be duds. I don't have time to do an example math problem, but anyone moderately adept at math would realize this is true.

Again though, this would require that the active manager have some system that's better than a completely random method of selecting stocks. If it is true, that no one really knows, then sure I'll concede just buying the index would mathematically have to be the best strategy.

I just like knowing an actual human is at the helm, such as like the volatile times we're in now. I just can't believe no one has any idea at all about what to do.

Many of us lack your confidence in the contribution of humans to these investment decisions. The evidence against humans being able to make useful predictions of stock market returns is extremely strong.

As for active investors almost all going to zero- we need not do thought experiments. We have at least 100 years of real life experience. Active investors consistently and reliably underperform the indexes over long term. But they do not all go to zero. If the math required them to do so, the entire industry should have collapsed long ago. Not because investors got smart and pulled their money. It should have collapsed because almost all the active investors should have gone bankrupt.

If something has not happened, that is pretty good evidence that it is not inevitable.
Of course, one could argue that it takes much longer than 100 years for this inescapable mathematical conclusion to prove out. In which case, who cares?

We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either |
--Swedroe |
We assume that markets are efficient, that prices are right |
--Fama

...
Quite frankly, I just like knowing an actual human is at the helm assessing whatever the current situation, such as like the volatile times we're in now. I just can't believe no one has any idea at all about what to do. For one, it's not possible for me to believe that because I can't unlearn my knowledge of Warren Buffett. Yeah, I know a famous guy we admire once said no one knows nothing, yet that same man paired down his stock holdings massively in the late 90s due to the current investing climate. And he was plenty older than I am now, when he did that.

There are many many well managed "active" mutual funds, that do just fine and have returns conservatively around the benchmark market returns. The issue, is when someone goes looking for something extra, how far from the norm does the manager have to deviate, how much risk is being assumed trying to capture something extra? There's also the issue of cost, how much is it worth to pay someone to conservatively deliver near-benchmark performance essentially doing what most would consider prudent investing, or how much are you willing to pay to have someone else make huge risky gambles that you don't understand (buy they assure you "trust me I know what I'm doing" - how do you tell the difference between real skill and a overconfident charlatans sales pitch) ?

"To achieve satisfactory investment results is easier than most people realize; to achieve superior results is harder than it looks." - Benjamin Graham

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

Yes, of course, but one could change some of the input parameters... Think to the sustained over-performance of Small-Cap Value compared to other strategies. The math theorem stays true, this is a more 'concentrated' investment, the standard-deviation was higher than the total-market returns, and yet it worked remarkably well in the past, definitely beating the market (crushing it, actually), and for more than a single winner. Why? Well, the arithmetic returns stayed consistently higher than the total market... Will that stay true in the future remains to be seen, but it certainly did work in the past, and not only in academic simulations, but also in real-life.

Anyway, it's worth understanding the abstract math model on its own merits (walk before run). Then figure when/where it's applicable to the messy concrete real world.

Yup, exactly, this is where the reasoning still nags me. It's an interesting discussion! Will reflect more upon it.

The math I said does NOT show that buying a concentrated part of the market and holding it is worse than holding the whole market. Obviously the concentrated bet could win. What is bad is repeatedly switching between different parts of the market. If you repeatedly do this randomly such that on average you held the market, then you almost certainly would have been better of holding the whole market.

Or if you repeatedly randomly switched between being 100% stock and 100% cash so that you were in each half the time, then you almost certainly would have been better of holding half of each the whole time.

So then there's no point to any of your math. Who is using a stock picking method of "randomly" switching between different parts of the market, or hiring someone who's doing that? I think it could have gone without saying that probably isn't a very good method of investing, and hopefully no one is doing that. If I'm going to hire an active manager, I'm going to pay a small premium to do it, and I'm going to have good reason to believe he has a better than average chance of beating a market index long-term, net of fees.

One would argue that, with respect to the realized risk adjusted returns, the stock picking by active investors is random. So it is reasonable to model their behavior as random deviations from the market. That does not mean they intend it to be random. They may well use highly systematic approaches. The systematic approaches do not increase risk adjusted returns. All they generate is (random) noise.

This does not lead one to the original conclusion that all but one active investor must go to zero, with the entire ownership of actively managed stocks vesting in one owner.

As for the talk of the "math" showing it. I have yet to see any math.

Note that the deviation of the arithmetic from the geometric mean applies just as much to the passive side of the market. It arises because of the way the two are calculated and the varying values of the stock market. It has nothing to do with active vs passive.

We don't know how to beat the market on a risk-adjusted basis, and we don't know anyone that does know either |
--Swedroe |
We assume that markets are efficient, that prices are right |
--Fama

I'm not sure what "compensates for the arithmetic/geometric issue" means , but arithmetic geometric mean inequality is a theorem, it's an absolute mathematical truth.

Yes, of course, but one could change some of the input parameters... Think to the sustained over-performance of Small-Cap Value compared to other strategies. The math theorem stays true, this is a more 'concentrated' investment, the standard-deviation was higher than the total-market returns, and yet it worked remarkably well in the past, definitely beating the market (crushing it, actually), and for more than a single winner. Why? Well, the arithmetic returns stayed consistently higher than the total market... Will that stay true in the future remains to be seen, but it certainly did work in the past, and not only in academic simulations, but also in real-life.

Anyway, it's worth understanding the abstract math model on its own merits (walk before run). Then figure when/where it's applicable to the messy concrete real world.

Yup, exactly, this is where the reasoning still nags me. It's an interesting discussion! Will reflect more upon it.

The math I said does NOT show that buying a concentrated part of the market and holding it is worse than holding the whole market. Obviously the concentrated bet could win. What is bad is repeatedly switching between different parts of the market. If you repeatedly do this randomly such that on average you held the market, then you almost certainly would have been better of holding the whole market.

Or if you repeatedly randomly switched between being 100% stock and 100% cash so that you were in each half the time, then you almost certainly would have been better of holding half of each the whole time.

So then there's no point to any of your math. Who is using a stock picking method of "randomly" switching between different parts of the market, or hiring someone who's doing that? I think it could have gone without saying that probably isn't a very good method of investing, and hopefully no one is doing that. If I'm going to hire an active manager, I'm going to pay a small premium to do it, and I'm going to have good reason to believe he has a better than average chance of beating a market index long-term, net of fees.

...
Quite frankly, I just like knowing an actual human is at the helm assessing whatever the current situation, such as like the volatile times we're in now. I just can't believe no one has any idea at all about what to do. For one, it's not possible for me to believe that because I can't unlearn my knowledge of Warren Buffett. Yeah, I know a famous guy we admire once said no one knows nothing, yet that same man paired down his stock holdings massively in the late 90s due to the current investing climate. And he was plenty older than I am now, when he did that.

There are many many well managed "active" mutual funds, that do just fine and have returns conservatively around the benchmark market returns. The issue, is when someone goes looking for something extra, how far from the norm does the manager have to deviate, how much risk is being assumed trying to capture something extra? There's also the issue of cost, how much is it worth to pay someone to conservatively deliver near-benchmark performance essentially doing what most would consider prudent investing, or how much are you willing to pay to have someone else make huge risky gambles that you don't understand (buy they assure you "trust me I know what I'm doing" - how do you tell the difference between real skill and a overconfident charlatans sales pitch) ?

I completely agree with all of that. One of my favorite examples of this is Vanguard Wellesley or Wellington. Neither stretch too far, and there's a small premium expense over index.

Bogleheads will take a nice step forward when we all figure out what is so bad about active is just when they're too expensive, which is still usually the case. Pure, 100%, EMH adherents I actually think are still in the minority. Specific example, I don't think many would dare to say Warren B just got lucky. A 100% EMH adherent would have to say that.